Colossal figure of merit and compelling HER catalytic activity of holey graphyne

Herein, we have conducted a comprehensive study to uncover the thermal transport properties and hydrogen evolution reaction catalytic activity of recently synthesized holey graphyne. Our findings disclose that holey graphyne has a direct bandgap of 1.00 eV using the HSE06 exchange–correlation functional. The absence of imaginary phonon frequencies in the phonon dispersion ensures its dynamic stability. The formation energy of holey graphyne turns out to be − 8.46 eV/atom, comparable to graphene (− 9.22 eV/atom) and h-BN (− 8.80 eV/atom). At 300 K, the Seebeck coefficient is as high as 700 μV/K at a carrier concentration of 1 × 1010 cm-2. The predicted room temperature lattice thermal conductivity (κl) of 29.3 W/mK is substantially lower than graphene (3000 W/mK) and fourfold smaller than C3N (128 W/mK). At around 335 nm thickness, the room temperature κl suppresses by 25%. The calculated p-type figure of merit (ZT) reaches a maximum of 1.50 at 300 K, higher than that of holey graphene (ZT = 1.13), γ-graphyne (ZT = 0.48), and pristine graphene (ZT = 0.55 × 10–3). It further scales up to 3.36 at 600 K. Such colossal ZT values make holey graphyne an appealing p-type thermoelectric material. Besides that, holey graphyne is a potential HER catalyst with a low overpotential of 0.20 eV, which further reduces to 0.03 eV at 2% compressive strain.

www.nature.com/scientificreports/ excellent mechanical, thermal, and dynamic stability. Unlike graphene, it is a direct bandgap semiconductor with high carrier mobility (promising for applications in optoelectronics) and possesses sp and sp 2 hybridized carbon atoms uniformly distributing the porous architectures (favorable for gas separation, water desalination, energy storage, and catalysis) 34 . Holey graphyne may also be considered an anchoring material in metal-sulfur batteries, like other materials with similar crystal structures previously researched for this purpose 35,36 . However, to the best of our knowledge, none of these applications of holey graphyne has been uncovered so far. Herein, we conducted a comprehensive study to explore its potential in thermoelectricity and H 2 production. By the presence of unique distinct bonding, κ l is anticipated to be lower in holey graphyne compared to the other flat 2D materials from the graphene family, resulting in an enhanced thermoelectric figure of merit. On the other hand, variation in bond charge density accompanied by a highly porous plane, which increases the number of reactive sites, makes it an excellent choice for catalysis.

Computational details
The Vienna Ab-initio Simulation Package (VASP) 37,38 is used to perform the density functional theory (DFT) calculations by employing the Perdew − Burke − Ernzerhof and Heyd − Scuseria − Ernzerhof (HSE06) hybrid exchange-correlation functionals 39,40 . A plane wave cutoff energy of 550 eV and a Γ-centred k-mesh of 9 × 9 × 1 (30 × 30 × 1) is used to sample the first Brillouin zone for self-consistent (non-self-consistent) calculations. The crystal is optimized until Hellmann-Feynman forces drop below 10 -4 eV/Å. A vacuum of 15 Å, perpendicular to the sheet, is taken to eliminate the interactions between adjacent layers. The structural formation energy is calculated using E form. = (E tot. − n.E C )/n formula, where E tot. and E C are the total energies of holey graphyne and an isolated C atom, respectively, and n is the total number of atoms in a unit cell. The phonon dispersion, thermal transport coefficients, and lattice thermal conductivity are calculated using the Hiphive 41 , BoltzTraP2 42 , and the ShengBTE 43 codes, respectively. This sophisticated method has been used to analyze the transport characteristics of numerous materials [44][45][46] . As inputs to ShengBTE, the 2nd and 3rd-order force constants are computed using a 3 × 3 × 1 supercell.

Results and discussions
Holey graphyne contains two rings comprised of six and eight carbon atoms, with a big pore, as shown in Fig. 1.
Each primitive cell has an optimized in-plane lattice constant of 10.84 Å, in agreement with the previous study 33 48 . It is worthwhile to point out that holey graphyne has been experimentally synthesized using the bottom-up approach 33 . Besides that, ab-initio molecular dynamics simulations have determined that holey graphyne is thermally stable even at higher temperatures of 900 K. 33  www.nature.com/scientificreports/ calculated direct bandgaps of 0.50 eV (PBE) and 1.0 eV (HSE06), which agrees with the previously calculated value and is also close to the experimental value of 1.10 eV. 33 .
The calculated phonon dispersion of holey graphyne has a total of 72 phonon modes without imaginary frequencies, assuring its dynamic stability (see Fig. 3). The out-of-plane flexural acoustic (ZA) phonons have the lowest frequency among the acoustic modes, followed by the in-plane transverse acoustic (TA) and longitudinal acoustic (LA) phonon modes. The first optical phonon mode has a noticeably small frequency of 2.55 THz, favoring the high phonon-phonon scattering owing to the coupling between ZA and the lowest optical phonons favors high phonon-phonon scattering 49,50 . Also, the less dispersive nature of optical phonons results in their small phonon group velocities. These distinct characteristics collectively contribute to low κ l , compared to other carbon allotropes. Figure 4 reveals the electronic transport coefficients with respect to varying carrier concentrations and temperatures. The electronic part of electrical (σ/τ) and thermal (κ e /τ) conductivities increases and the Seebeck coefficient (S) declines linearly with carrier concentration 51 . The room temperature |S| (300 µV/K at 1 × 10 12 cm −2 ) is approximately 6 times greater than that of graphene (~ 50 µV/K) at the same carrier concentration 52 . The p-type (n-type) |S| turns out to be 114 µV/K (110 µV/K) even at the highest considered carrier concentration (2 × 10 13 cm −2 ) and at 600 K. The p-type S 2 σ/τ rises with growing carrier concentration until it approaches 5 × 10 11 W/mK 2 s (9 × 10 11 W/mK 2 s) at 300 K (600 K) and then drops as the doping concentration elevates. Such a remarkably high S and S 2 σ/τ suggest that investigating holey graphyne for thermoelectrics is worthwhile. It is further notable that identical dispersions of the valence band maxima and the conduction band minima result in similar variance in p-type and n-type electronic transport coefficients.
Further, we have looked into how lattice vibrations affect heat conduction in holey graphyne (see Fig. 5a). The calculated room temperature κ l turns out to be 29.3 W/mK, lower than graphene (3000 W/mK) 53 , γ-graphyne (76.4 W/mK) 54 , C 3 N (128 W/mK) 53 , and C 2 N (82.22 W/mK) 55 . The calculated κ l at 300 K differs by 0.01% from the results obtained using 20 × 20 × 1 q-mesh (i.e., 29.4 W/mK), leaving no relevant effect on our conclusion. The atom displacement to the ninth nearest neighbors yields κ l of 28.05 W/mK, which deviates by 4% from the value obtained by displacing atoms up to the tenth nearest neighbors. The ultralow κ l of holey graphyne, which is appealing from the thermoelectric perspective, is attributed to its bonds being less stiff than those of the abovementioned flat materials. The C-C bond stiffness is determined by the spring constant (K) of holey graphyne, which is calculated as the trace of the harmonic force constant tensor between the closest adjacent atoms. It is written as K = xx CC + yy CC + zz CC , where � αα MX is the second derivate of energy with respect to displacement of atoms along the Cartesian axis α. The bond stiffness of d1, d2, d3, and d4 are 87 eV/Å 2 , 36 eV/Å 2 , 44 eV/Å 2 , and 31 eV/Å 2 , which are substantially smaller than that of graphene (10,105 eV/Å 2 ). Such a bond feature inhibits heat transfer via lattice vibrations, leading to low κ l values. Figure 5a shows that as temperature increases κ l decreases due to the pronounced phonon-phonon scattering and follows the relation κ l ∝ 1/T. www.nature.com/scientificreports/ Nanostructuring of materials, where materials are composed of nanometer size grains and further with nanoscale internal structures 56 , can reduce κ l without affecting σ. To better comprehend the scope of phonon engineering, the cumulative lattice thermal conductivity (κ cl ) as a function of the phonon mean free path (MFP) is investigated and presented in Fig. 5b. As MFP decreases, the phonons scattering increases reducing the heat transfer. The contribution of phonons with different MFPs to lattice thermal conductivity is studied by calculating κ cl , thereby deducing phonons most relevant to thermal conduction 43 . At 300 K (600 K) 75% of κ l is by phonons having MFP 335 nm (110 nm), which implies reducing κ l through nanostructuring is a viable strategy for holey graphyne. In other words, a sample size of 335 nm (110 nm) could help reduce the inherited value of κ l at 300 K (600 K) by one-fourth. κ cl increases (decreases) with rising MFP (temperature) and shows a plateau above 20,092 nm (7924 nm) at 300 K (600 K). A lower plateau at 600 K is due to the stronger phonon scattering at elevated temperatures 10 . Given that phonon transports are predominantly dependent on phonon scattering rates and phonon group velocities, our calculated results for acoustic and optical phonon modes are illustrated in Fig. 5c,d. The highest scattering rate for acoustic phonons at room temperature is 2.74 ps −1 , comparatively higher  www.nature.com/scientificreports/ than C 3 N monolayer (2 ps −1 ) 53 . It is due to the coupling of acoustic and optical phonon modes, which leads to increased scattering rates of acoustic phonons, leading to a considerable reduction of κ l 50 . The phonon group velocities are calculated to validate the above analysis and presented in Fig. 5d. The highest phonon group velocity of acoustic phonon mode at room temperature is 13.9 km/s, much lower than that of graphene (~ 22 km/s) 57 and nitrogenated holey graphene (18.48 km/s) 58 . Such low group velocity is a consequence of flat phonon modes (see Fig. 3) 50 . Thus, smaller κ l of holey graphyne arises from low phonon group velocity and high scattering rates.
The figure of merit (ZT) depends on the relaxation time (τ) and is vital for showcasing the potential of a material for its thermoelectric applications. In this work, we have adapted the value for τ from the deformation potential theory used in the previous study 33 . The relation τ T = 300*τ 300 /T is engaged to yield its value at 600 K. The values of τ for holes (electrons) turn out to be 3.27 ps (1.16 ps) and 1.64 ps (0.58 ps) at 300 K and 600 K, respectively. The calculated ZT in Fig. 6 follows an upward trend similar to S 2 σ/τ. The p-type ZT is higher than the n-type and is counter to the trend seen in electronic transport coefficients (see Fig. 4). It is a result of the fact that holes and electrons have different relaxation times. The relation ZT = S 2 T/(κ e /σ + κ l /σ), where κ l /σ is influenced by τ, explains that larger τ values result in greater ZT. The ZT secures peak values of 3.36 (1.50) and 1.82 (0.71) at 600 K (300 K) for p-type and n-type dopings, respectively. In the previous study 33 , ab-initio molecular dynamics simulations determined that holey graphyne is thermally stable even at higher temperatures of 900 K, as established based on the ab initio molecular dynamics simulations. The room temperature p-type ZT of holey graphyne is higher than that of holey graphene (1.13) 59 and significantly higher than pristine graphene (0.55 × 10 -3 ) 60 and γ-graphyne (0.48) 54 .
Since active and cost-effective metal-free catalysts, as opposed to costly Pt-based catalysts, for HER are necessary to attain sustainable energy systems. We have also uncovered the response of holey graphyne to HER. As the standard descriptor for HER activity is the Gibbs free energy (ΔG H ), which is derived from the hydrogen adsorption energy. In the first instance, the H atom is adsorbed on all the possible 9 adsorption sites (s1-s9 in Fig. 7a). The H atom finds the s2 site the most favorable. The corresponding ΔG H of s2 site is 0.20 eV, which is much smaller than that of graphene (1.41 eV) 61 , phosphorene (1.25 eV) 62 , and C 3 N 4 (0.58 eV) 63 and comparable to biphenylene (0.29 eV) 61 , see Fig. 7b. As a matter of fact, ΔG H = 0 stands out as an optimal value for HER. However, a value of |ΔG H |< 0.2 eV signifies the better catalytic performance of materials for HER activity 64 . We further engaged strain engineering to evaluate its impact on ΔG H . It is observed that 2% compressive strain improves the catalytic performance of holey graphyne by reducing ΔG H to 0.03 eV. Moreover, the-ab initio molecular dynamics simulation assures the thermal stability of hydrogen adsorbed holey graphyne concerning the low energy fluctuations alongside the intact H and the sheet structure after a time of 8 ps (see Fig. 7c). To address the photocatalytic performance of holey graphyne, the positions of valence band maxima (VBM) and conduction band minima (CBM) relative to the vacuum level along with H + /H 2 reduction and (O 2 /H 2 O) oxidation potentials for water splitting are presented in Fig. 7d. It is evident that CBM is more positive than the H + /H 2 potential, suggesting that holey graphyne is a potential material for photocatalytic hydrogen production. In contrast, the VBM is higher than the O 2 /H 2 O oxidation potential; hence, the holey graphene is unsuitable for oxidizing H 2 O to O 2 . However, an appropriate band engineering to shift the VBM downwards (e.g., doping, application of an external bias, heterojunction, etc.) may enable the water splitting into H 2 and O 2 .

Conclusions
Using the first-principles computations, we investigated the thermal transport properties and HER catalytic activity of holey graphyne, a semiconducting material with a direct bandgap of 1.0 eV. The closely packed phonons display positive frequencies across the entire Brillouin zone, assuring the dynamic stability of holey graphyne. The room temperature |S| is as high as 300 μV/K at 1 × 10 12 cm −2 (nearly sixfold than graphene). Flexural acoustic phonons couple with the lowest optical phonon mode to enhance phonon-phonon scattering and eventually decrease κ l . The estimated room temperature κ l of 29.3 W/mK is ultralow compared to that of graphene (3000 W/mK), attributed to its low bond stiffness. With the aid of nanostructure engineering, κ l further reduces by a factor of four for a crystal size of around 335 nm (110 nm) at 300 K (600 K). The room temperature p-type ZT achieves a maximum value of 1.50, largely greater than graphene (0.55 × 10 -3 ), demonstrating its superior thermoelectric performance. Lastly, holey graphyne having low overpotential and more positive CBM than the H + /H 2 potential, is also capable of catalyzing HER.

Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.